The Strengthening and Weakening Instrument: Comparability of Topological Spaces

B. Dvalishvili

Along with noncomparable topologies, the paper concentrates on situations, where in a bitopological space one topology is finer than the other, which is frequently encountered in applications. In this context, different families of sets are considered and the bitopological modification of the Cantor-Bendixson theorem is proved. The three operators are defined, which characterize the degrees of nearness of the four boundaries of any set, tangency of topologies, S-, C- and N-relations, and thus make it possible to compare small inductive dimensions at some special point. Furthermore, different properties of pair wise small and pair wise large inductive dimensions are studied. In the final part, the conditions are given, under which a bitopological space preserves the property to be an (i,j)-Baire space to the image and preimage. Relations between pair wise small and large inductive dimensions of the domain and the range of a d-closed and d-continuous function are investigated.